Binomial edge ideals and conditional independence statements
Abstract
We introduce binomial edge ideals attached to a simple graph and study their algebraic properties. We characterize those graphs for which the quadratic generators form a Gr\"obner basis in a lexicographic order induced by a vertex labeling. Such graphs are chordal and claw-free. We give a reduced squarefree Gr\"obner basis for general . It follows that all binomial edge ideals are radical ideals. Their minimal primes can be characterized by particular subsets of the vertices of . We provide sufficient conditions for Cohen--Macaulayness for closed and nonclosed graphs. Binomial edge ideals arise naturally in the study of conditional independence ideals. Our results apply for the class of conditional independence ideals where a fixed binary variable is independent of a collection of other variables, given the remaining ones. In this case the primary decomposition has a natural statistical interpretation
Cite
@article{arxiv.0909.4717,
title = {Binomial edge ideals and conditional independence statements},
author = {Juergen Herzog and Takayuki Hibi and Freyja Hreinsdottir and Thomas Kahle and Johannes Rauh},
journal= {arXiv preprint arXiv:0909.4717},
year = {2009}
}